ABSTRACT

In this paper, we consider the two-sided first exit problem for jump diffusion processes having jumps with rational Laplace transforms. We investigate the probabilistic property of conditional memorylessness, and drive the joint distribution of the first exit time from an interval and the overshoot over the boundary at the exit time.

1. Introduction

Consider the following jump diffusion process

(1.1)

where the constant is the starting point of, and represent the drift and the volatility of the diffusion part, respectively, is a standard Brownian motion with, is a Poisson process with rate, and the jumps sizes are assumed to be i.i.d. real valued random variables with common density. Moreover, it is assumed that the random processes, and random variables are mutually independent. In this paper we are interested in the density of following type

(1.2)

where, , , and that, for all. Moreover,

Define to be the first exit time of to two flat barriers and , i.e.

Recently, one-sided and two-sided first exit problems for processes with two-sided jumps have attracted a lot of attentions in applied probability (see [1-7]). For example, Perry and Stadje [1] studied two-sided first exit time for processes with two-sided exponential jumps; Kou and Wang [2] studied the one-sided first passage times for a jump diffusion process with exponential positive and negative jumps. Cai [3] investigated the first passage time of a hyper-exponential jump diffusion process. Cai et al. [4] discussed the first passage time to two barriers of a hyper-exponential jump diffusion process. Closed form expressions are obtained in Kadankova and Veraverbeke [5] for the integral transforms of the joint distribution of the first exit time from an interval and the value of the overshoot through boundaries at the exit time for the Poisson process with an exponential component. For some related works, see Perry et al. [8], Cai and Kou [9], Lewis and Mordecki [10] and the references therein.

Motivated by works mentioned above, the main objective of this paper is to study the first exit time of the process (1.1) with jump density (1.2) from an interval and the overshoot over the boundary at the exit time. In Section 2, we study the roots of the generalized Lundberg equation and conditional memory lessness. The main results of this paper are given in Section 3.

2. Preliminary Results

It is easy to see that the infinitesimal generator of is given by

for any twice continuously differentiable function and the Lévy exponent of is given by

By analytic continuation, the function can be extended to the complex plane except at finitely many poles. In the following, we consider the resulting extension of, i.e.,

Let us denote and.

In [11], Kuznetsov has studied the roots of the equation. However, for this particular Lévy process, we will give another simple proof for the roots of this equation.

Lemma 2.1. For fix, the generalized CramérLundberg equation

has complex roots with for and with for.

Proof. Let

Firstly, we prove that for given, has roots with negative real parts. Set

withwhere is an arbitrary positive constant. Applying Rouchés theorem on the semi-circle, consisting of the imaginary axis running from to and with radius running clockwise from to. We let and denote by the limiting semi-circle. It is known that both and are analytic in. We want to show that

Notice that for, and

is bounded for. Hence, for,

on the boundary of the half circle in. For, we have (see Lewis and Mordecki [10]). On the other hand,

Thus we have. Since

has roots with negative real parts, so equation has roots with negative real parts. Similarly, we can prove has roots with positive real parts.

In the rest of this paper, we assume all the roots of equation are distinct and denote , for notational simplicity, and denote (or in the sequel) representing the expectation (or probability) when starts from. We denote a sequence of events

,

= {: crosses at time by the th phase of th positive jump whose parameter is },

= {: crosses at time by the th phase of th negative jump whose parameter is }

for, , , , and.

Theorem 2.2. For any, we have

(2.1)

(2.2)

Furthermore, conditional on , the stopping time is independent of the overshoot (the undershoot). More precisely, for any, we have

(2.3)

(2.4)

Proof. Firstly, we prove (2.1) and (2.3). It suffices to show

(2.5)

since (2.1) can be obtained by letting in (2.5) and then dividing both sides of the resulting equation by. It is known that an Erlang(n) random variable can be expressed as an independent sum of exponential random variables with same parameters. Let the independent exponentially distributed random variables with parameter. Denote by the arrival times of the Poisson process, and let be the field generated by process,. It follows that

With, we have

Thus we have

(2.2) and (2.4) can be obtained similarly. This completes the proof.

The following results are immediate consequences of Theorem 2.2.

Corollary 2.3. For, , , , , , we have

Corollary 2.4. For any, we have

where

for, , , .

Corollary 2.5. For, , , , we have

Remark 2.6. When, , (2.1) and (2.3) reduce to Equations (8) and (9) of Cai [3], respectively.

3. Main Results

In this section, we study the distribution of the first exit problem to two barriers. We first define three vectors:

where

Let

Define a matrix .

Theorem 3.1. Consider any nonnegative measurable function such that and for, , ,. For any and, we have

(3.1)

where satisfies

(3.2)

Moreover, when is a non-singular matrix, is the unique solution of (3.2), i.e.,

(3.3)

Proof. By the law of total probability, we have

It follows from Corollary 2.4, for, , , , we have

Combining these equations, we get

The expressions for, , and can be determined as follows. Let denote the set of functions

such that is twice continuously differentiable and bounded for with and bounded for. By applying Itô formula to the process, we have that for and,

where is a martingale with. Note that we have as.

For any, we can easily obtain from the above equation that

where the last term of the above equation is a mean-0 martingale. This implies that

(3.4)

By simple calculation, the function with and satisfies for. It follows from (3.4) that the process

is a martingale. Then

(3.5)

Setting for and for in (3.5), we have the following linear equations:

and

Then the vector satisfies and we have (3.1). If is non-singular, we have . This completes the proof.

Corollary 3.2. For any

, we have

(3.6)

where

and

Remark 3.3. When, , (3.1) and (3.6) reduce to equation (6) and (15) of [4], respectively.

From Theorem 3.1, choosing to be,

, , , , and respectively, we can obtain the following corollaries.

Corollary 3.4. 1) For any, we have

(3.7)

where

is determined by the linear system. Here

2) For any, we have

(3.8)

where

is determined by the linear system. Here

Corollary 3.5. 1) For and any, , we have

(3.9)

where

is determined by the linear system. Here

2) For and any, , we have

(3.10)

where

is determined by the linear system. Here

Note that the difference of and is exactly . Thus we obtain the following results.

Corollary 3.6. 1) For, and for any, we have

(3.11)

where

is determined by the linear system. Here

2) For and any, , we have

(3.12)

where

is determined by the linear system. Here

To end the paper, we give an example.

Example 3.7. When,

and

, the equation has real roots:, , and . Let

Denote by

Then we have

where

We define (, ,) and (, ,) as follows: let (, ,) be obtained from (, ,) by changing to in (,); let (, ,) be obtained from (, ,) by changing to in (, ,).

• If, then we have

where

•

• If, then we have

where

•

• If, then we have

where

•

• If, , then we have

•

• If, , then we have

where

•

• If, then we have

where

•

• If, then we have

where

When, we have

Therefore, we have

These results are all consistent with that of Theorem 3.1 of Kou and Wang [2] for the one-sided exit problem of the doubly exponential jump diffusion process.

REFERENCES

NOTES

^*This work was supported by the National Natural Science Foundation of China (No. 11171179) and Natural Science Foundation of Shandong Province (No. ZR2010AQ015).

^#Corresponding author.